Lemma 21.13.3. Let \mathcal{C} be a site. Let K be a sheaf of sets on \mathcal{C}. Consider the morphism of topoi j : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}), see Sites, Lemma 7.30.3. Then j^{-1} preserves injectives and H^ p(K, \mathcal{F}) = H^ p(\mathcal{C}/K, j^{-1}\mathcal{F}) for any abelian sheaf \mathcal{F} on \mathcal{C}.
Proof. By Sites, Lemmas 7.30.1 and 7.30.3 the morphism of topoi j is equivalent to a localization. Hence this follows from Lemma 21.7.1. \square
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